Integrand size = 31, antiderivative size = 55 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \] Output:
1/4*a*f^3*(e*x+d)^4/e+1/6*b*f^3*(e*x+d)^6/e+1/8*c*f^3*(e*x+d)^8/e
Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(55)=110\).
Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.80 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=f^3 \left (d^3 \left (a+b d^2+c d^4\right ) x+\frac {1}{2} d^2 \left (3 a+5 b d^2+7 c d^4\right ) e x^2+\frac {1}{3} d \left (3 a+10 b d^2+21 c d^4\right ) e^2 x^3+\frac {1}{4} \left (a+10 b d^2+35 c d^4\right ) e^3 x^4+d \left (b+7 c d^2\right ) e^4 x^5+\frac {1}{6} \left (b+21 c d^2\right ) e^5 x^6+c d e^6 x^7+\frac {1}{8} c e^7 x^8\right ) \] Input:
Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]
Output:
f^3*(d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3*a + 10*b*d^2 + 21*c*d^4)*e^2*x^3)/3 + ((a + 10*b*d^2 + 35*c*d^4)*e^ 3*x^4)/4 + d*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^6* x^7 + (c*e^7*x^8)/8)
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1462, 1433, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx\) |
\(\Big \downarrow \) 1462 |
\(\displaystyle \frac {f^3 \int (d+e x)^3 \left (c (d+e x)^4+b (d+e x)^2+a\right )d(d+e x)}{e}\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \frac {f^3 \int \left (c (d+e x)^7+b (d+e x)^5+a (d+e x)^3\right )d(d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {f^3 \left (\frac {1}{4} a (d+e x)^4+\frac {1}{6} b (d+e x)^6+\frac {1}{8} c (d+e x)^8\right )}{e}\) |
Input:
Int[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]
Output:
(f^3*((a*(d + e*x)^4)/4 + (b*(d + e*x)^6)/6 + (c*(d + e*x)^8)/8))/e
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Si mp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p , x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(49)=98\).
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.25
method | result | size |
gosper | \(\frac {f^{3} x \left (3 c \,e^{7} x^{7}+24 c d \,e^{6} x^{6}+84 x^{5} c \,d^{2} e^{5}+168 c \,d^{3} e^{4} x^{4}+4 x^{5} b \,e^{5}+210 x^{3} c \,d^{4} e^{3}+24 b d \,e^{4} x^{4}+168 x^{2} c \,d^{5} e^{2}+60 x^{3} b \,d^{2} e^{3}+84 x c \,d^{6} e +80 x^{2} b \,d^{3} e^{2}+24 c \,d^{7}+6 x^{3} a \,e^{3}+60 x b \,d^{4} e +24 x^{2} a d \,e^{2}+24 b \,d^{5}+36 x a \,d^{2} e +24 a \,d^{3}\right )}{24}\) | \(179\) |
norman | \(\left (\frac {7}{2} d^{2} f^{3} e^{5} c +\frac {1}{6} b \,e^{5} f^{3}\right ) x^{6}+\left (7 c \,d^{5} e^{2} f^{3}+\frac {10}{3} b \,d^{3} e^{2} f^{3}+a d \,e^{2} f^{3}\right ) x^{3}+\left (\frac {7}{2} c \,d^{6} e \,f^{3}+\frac {5}{2} b \,d^{4} e \,f^{3}+\frac {3}{2} a \,d^{2} e \,f^{3}\right ) x^{2}+\left (\frac {35}{4} d^{4} f^{3} c \,e^{3}+\frac {5}{2} b \,d^{2} e^{3} f^{3}+\frac {1}{4} a \,e^{3} f^{3}\right ) x^{4}+\left (7 d^{3} f^{3} c \,e^{4}+b d \,e^{4} f^{3}\right ) x^{5}+\left (c \,d^{7} f^{3}+b \,d^{5} f^{3}+a \,d^{3} f^{3}\right ) x +d \,f^{3} e^{6} c \,x^{7}+\frac {e^{7} f^{3} c \,x^{8}}{8}\) | \(216\) |
risch | \(\frac {1}{8} e^{7} f^{3} c \,x^{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {7}{2} f^{3} x^{6} c \,d^{2} e^{5}+\frac {1}{6} f^{3} x^{6} b \,e^{5}+7 f^{3} c \,d^{3} e^{4} x^{5}+f^{3} b d \,e^{4} x^{5}+\frac {35}{4} f^{3} x^{4} c \,d^{4} e^{3}+\frac {5}{2} f^{3} x^{4} b \,d^{2} e^{3}+\frac {1}{4} f^{3} x^{4} a \,e^{3}+7 f^{3} x^{3} c \,d^{5} e^{2}+\frac {10}{3} f^{3} x^{3} b \,d^{3} e^{2}+f^{3} x^{3} a d \,e^{2}+\frac {7}{2} f^{3} x^{2} c \,d^{6} e +\frac {5}{2} f^{3} x^{2} b \,d^{4} e +\frac {3}{2} f^{3} x^{2} a \,d^{2} e +f^{3} c \,d^{7} x +f^{3} b \,d^{5} x +f^{3} a \,d^{3} x\) | \(230\) |
parallelrisch | \(\frac {1}{8} e^{7} f^{3} c \,x^{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {7}{2} f^{3} x^{6} c \,d^{2} e^{5}+\frac {1}{6} f^{3} x^{6} b \,e^{5}+7 f^{3} c \,d^{3} e^{4} x^{5}+f^{3} b d \,e^{4} x^{5}+\frac {35}{4} f^{3} x^{4} c \,d^{4} e^{3}+\frac {5}{2} f^{3} x^{4} b \,d^{2} e^{3}+\frac {1}{4} f^{3} x^{4} a \,e^{3}+7 f^{3} x^{3} c \,d^{5} e^{2}+\frac {10}{3} f^{3} x^{3} b \,d^{3} e^{2}+f^{3} x^{3} a d \,e^{2}+\frac {7}{2} f^{3} x^{2} c \,d^{6} e +\frac {5}{2} f^{3} x^{2} b \,d^{4} e +\frac {3}{2} f^{3} x^{2} a \,d^{2} e +f^{3} c \,d^{7} x +f^{3} b \,d^{5} x +f^{3} a \,d^{3} x\) | \(230\) |
orering | \(\frac {x \left (3 c \,e^{7} x^{7}+24 c d \,e^{6} x^{6}+84 x^{5} c \,d^{2} e^{5}+168 c \,d^{3} e^{4} x^{4}+4 x^{5} b \,e^{5}+210 x^{3} c \,d^{4} e^{3}+24 b d \,e^{4} x^{4}+168 x^{2} c \,d^{5} e^{2}+60 x^{3} b \,d^{2} e^{3}+84 x c \,d^{6} e +80 x^{2} b \,d^{3} e^{2}+24 c \,d^{7}+6 x^{3} a \,e^{3}+60 x b \,d^{4} e +24 x^{2} a d \,e^{2}+24 b \,d^{5}+36 x a \,d^{2} e +24 a \,d^{3}\right ) \left (e f x +d f \right )^{3} \left (a +b \left (e x +d \right )^{2}+c \left (e x +d \right )^{4}\right )}{24 \left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right ) \left (e x +d \right )^{3}}\) | \(279\) |
default | \(\frac {e^{7} f^{3} c \,x^{8}}{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {\left (15 d^{2} f^{3} e^{5} c +e^{3} f^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )\right ) x^{6}}{6}+\frac {\left (13 d^{3} f^{3} c \,e^{4}+3 d \,f^{3} e^{2} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+e^{3} f^{3} \left (4 c \,d^{3} e +2 b d e \right )\right ) x^{5}}{5}+\frac {\left (4 d^{4} f^{3} c \,e^{3}+3 d^{2} f^{3} e \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d \,f^{3} e^{2} \left (4 c \,d^{3} e +2 b d e \right )+e^{3} f^{3} \left (c \,d^{4}+b \,d^{2}+a \right )\right ) x^{4}}{4}+\frac {\left (d^{3} f^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d^{2} f^{3} e \left (4 c \,d^{3} e +2 b d e \right )+3 d \,f^{3} e^{2} \left (c \,d^{4}+b \,d^{2}+a \right )\right ) x^{3}}{3}+\frac {\left (d^{3} f^{3} \left (4 c \,d^{3} e +2 b d e \right )+3 d^{2} f^{3} e \left (c \,d^{4}+b \,d^{2}+a \right )\right ) x^{2}}{2}+x \,d^{3} f^{3} \left (c \,d^{4}+b \,d^{2}+a \right )\) | \(349\) |
Input:
int((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)
Output:
1/24*f^3*x*(3*c*e^7*x^7+24*c*d*e^6*x^6+84*c*d^2*e^5*x^5+168*c*d^3*e^4*x^4+ 4*b*e^5*x^5+210*c*d^4*e^3*x^3+24*b*d*e^4*x^4+168*c*d^5*e^2*x^2+60*b*d^2*e^ 3*x^3+84*c*d^6*e*x+80*b*d^3*e^2*x^2+24*c*d^7+6*a*e^3*x^3+60*b*d^4*e*x+24*a *d*e^2*x^2+24*b*d^5+36*a*d^2*e*x+24*a*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (49) = 98\).
Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.02 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \] Input:
integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")
Output:
1/8*c*e^7*f^3*x^8 + c*d*e^6*f^3*x^7 + 1/6*(21*c*d^2 + b)*e^5*f^3*x^6 + (7* c*d^3 + b*d)*e^4*f^3*x^5 + 1/4*(35*c*d^4 + 10*b*d^2 + a)*e^3*f^3*x^4 + 1/3 *(21*c*d^5 + 10*b*d^3 + 3*a*d)*e^2*f^3*x^3 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*a* d^2)*e*f^3*x^2 + (c*d^7 + b*d^5 + a*d^3)*f^3*x
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (44) = 88\).
Time = 0.04 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.36 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=c d e^{6} f^{3} x^{7} + \frac {c e^{7} f^{3} x^{8}}{8} + x^{6} \left (\frac {b e^{5} f^{3}}{6} + \frac {7 c d^{2} e^{5} f^{3}}{2}\right ) + x^{5} \left (b d e^{4} f^{3} + 7 c d^{3} e^{4} f^{3}\right ) + x^{4} \left (\frac {a e^{3} f^{3}}{4} + \frac {5 b d^{2} e^{3} f^{3}}{2} + \frac {35 c d^{4} e^{3} f^{3}}{4}\right ) + x^{3} \left (a d e^{2} f^{3} + \frac {10 b d^{3} e^{2} f^{3}}{3} + 7 c d^{5} e^{2} f^{3}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e f^{3}}{2} + \frac {5 b d^{4} e f^{3}}{2} + \frac {7 c d^{6} e f^{3}}{2}\right ) + x \left (a d^{3} f^{3} + b d^{5} f^{3} + c d^{7} f^{3}\right ) \] Input:
integrate((e*f*x+d*f)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4),x)
Output:
c*d*e**6*f**3*x**7 + c*e**7*f**3*x**8/8 + x**6*(b*e**5*f**3/6 + 7*c*d**2*e **5*f**3/2) + x**5*(b*d*e**4*f**3 + 7*c*d**3*e**4*f**3) + x**4*(a*e**3*f** 3/4 + 5*b*d**2*e**3*f**3/2 + 35*c*d**4*e**3*f**3/4) + x**3*(a*d*e**2*f**3 + 10*b*d**3*e**2*f**3/3 + 7*c*d**5*e**2*f**3) + x**2*(3*a*d**2*e*f**3/2 + 5*b*d**4*e*f**3/2 + 7*c*d**6*e*f**3/2) + x*(a*d**3*f**3 + b*d**5*f**3 + c* d**7*f**3)
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (49) = 98\).
Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.02 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \] Input:
integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")
Output:
1/8*c*e^7*f^3*x^8 + c*d*e^6*f^3*x^7 + 1/6*(21*c*d^2 + b)*e^5*f^3*x^6 + (7* c*d^3 + b*d)*e^4*f^3*x^5 + 1/4*(35*c*d^4 + 10*b*d^2 + a)*e^3*f^3*x^4 + 1/3 *(21*c*d^5 + 10*b*d^3 + 3*a*d)*e^2*f^3*x^3 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*a* d^2)*e*f^3*x^2 + (c*d^7 + b*d^5 + a*d^3)*f^3*x
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (49) = 98\).
Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.71 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} c d^{6} f^{2} + \frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} b d^{4} f^{2} + \frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} a d^{2} f^{2} + \frac {18 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} c d^{4} e f^{2} + 12 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} c d^{2} e^{2} f + 3 \, {\left (e f x^{2} + 2 \, d f x\right )}^{4} c e^{3} + 12 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} b d^{2} e f^{2} + 4 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} b e^{2} f + 6 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} a e f^{2}}{24 \, f} \] Input:
integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")
Output:
1/2*(e*f*x^2 + 2*d*f*x)*c*d^6*f^2 + 1/2*(e*f*x^2 + 2*d*f*x)*b*d^4*f^2 + 1/ 2*(e*f*x^2 + 2*d*f*x)*a*d^2*f^2 + 1/24*(18*(e*f*x^2 + 2*d*f*x)^2*c*d^4*e*f ^2 + 12*(e*f*x^2 + 2*d*f*x)^3*c*d^2*e^2*f + 3*(e*f*x^2 + 2*d*f*x)^4*c*e^3 + 12*(e*f*x^2 + 2*d*f*x)^2*b*d^2*e*f^2 + 4*(e*f*x^2 + 2*d*f*x)^3*b*e^2*f + 6*(e*f*x^2 + 2*d*f*x)^2*a*e*f^2)/f
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.98 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {e^5\,f^3\,x^6\,\left (21\,c\,d^2+b\right )}{6}+\frac {c\,e^7\,f^3\,x^8}{8}+d^3\,f^3\,x\,\left (c\,d^4+b\,d^2+a\right )+\frac {e^3\,f^3\,x^4\,\left (35\,c\,d^4+10\,b\,d^2+a\right )}{4}+\frac {d^2\,e\,f^3\,x^2\,\left (7\,c\,d^4+5\,b\,d^2+3\,a\right )}{2}+\frac {d\,e^2\,f^3\,x^3\,\left (21\,c\,d^4+10\,b\,d^2+3\,a\right )}{3}+d\,e^4\,f^3\,x^5\,\left (7\,c\,d^2+b\right )+c\,d\,e^6\,f^3\,x^7 \] Input:
int((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x)
Output:
(e^5*f^3*x^6*(b + 21*c*d^2))/6 + (c*e^7*f^3*x^8)/8 + d^3*f^3*x*(a + b*d^2 + c*d^4) + (e^3*f^3*x^4*(a + 10*b*d^2 + 35*c*d^4))/4 + (d^2*e*f^3*x^2*(3*a + 5*b*d^2 + 7*c*d^4))/2 + (d*e^2*f^3*x^3*(3*a + 10*b*d^2 + 21*c*d^4))/3 + d*e^4*f^3*x^5*(b + 7*c*d^2) + c*d*e^6*f^3*x^7
Time = 0.22 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.24 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {f^{3} x \left (3 c \,e^{7} x^{7}+24 c d \,e^{6} x^{6}+84 c \,d^{2} e^{5} x^{5}+168 c \,d^{3} e^{4} x^{4}+4 b \,e^{5} x^{5}+210 c \,d^{4} e^{3} x^{3}+24 b d \,e^{4} x^{4}+168 c \,d^{5} e^{2} x^{2}+60 b \,d^{2} e^{3} x^{3}+84 c \,d^{6} e x +80 b \,d^{3} e^{2} x^{2}+24 c \,d^{7}+6 a \,e^{3} x^{3}+60 b \,d^{4} e x +24 a d \,e^{2} x^{2}+24 b \,d^{5}+36 a \,d^{2} e x +24 a \,d^{3}\right )}{24} \] Input:
int((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x)
Output:
(f**3*x*(24*a*d**3 + 36*a*d**2*e*x + 24*a*d*e**2*x**2 + 6*a*e**3*x**3 + 24 *b*d**5 + 60*b*d**4*e*x + 80*b*d**3*e**2*x**2 + 60*b*d**2*e**3*x**3 + 24*b *d*e**4*x**4 + 4*b*e**5*x**5 + 24*c*d**7 + 84*c*d**6*e*x + 168*c*d**5*e**2 *x**2 + 210*c*d**4*e**3*x**3 + 168*c*d**3*e**4*x**4 + 84*c*d**2*e**5*x**5 + 24*c*d*e**6*x**6 + 3*c*e**7*x**7))/24